Here is a notion of set-theoretic truth that does not require inaccessibles.
Let's expand the language of set theory to include a constant /kappa, and the axiom scheme
phi iff V_/kappa satisfies phi
, for ALL sentences phi in the original language (not including the symbol /kappa)..
/kappa shall henceforth be referred to as an "infallible cardinal" (being a Catholic, I find this a congenial concept).
If an infallible cardinal exists, it provides a nice semantics for set theory, but in order to avoid circularity we need to say something else about infallible cardinals. Here are some questions:
1) If an infallible cardinal exists, what non-ZFC-provable sentences in the language of set theory must be true?
2) If no infallible cardinal exists, what consequences follow?
3) Suppose the Universe is Mahlo, and there is a stationary class of inaccessibles such every sentence is either eventually true or eventually false in the corresponding set of V_j. What is the relation between this assumption, and the assumption that an infallible cardinal exists? (Can a college of cardinals always identify an infallible one?)
Let's try to formalize this a little more:
Definition:
A "college of cardinals" is a Mahlo class of inaccessibles such that every sentence in the language of set theory is either eventually true or eventually false in the corresponding class of standard models V_j.
A "University" is a college which is a proper class.
A college "elects" a member k if every phi satisfied by V_k is eventually true in the class of standard models V_j for j in the college.
Call the elected cardinal a Pope. It should be provable that every college can elect a Pope. The real question is
Definition (requires a semantics for the language of set theory). k is an "infallible" cardinal if V_k satisfies exactly the true sentences of set theory.
Proposition: If a college of cardinals is actually a University, it elects an infallible Pope.
Question: Is there a reasonable semantics for Set Theory under which the above Proposition holds?
-- Joe Shipman